Collisional Effects on the Drift Hole Dynamics in Magnetically Confined Plasmas

Abstract

The effect of collisions on the dynamics of phase space structures in inhomogeneous magnetized plasmas is analyzed. A drift hole is used as an example, and subcritical bifurcation associated with a drift hole is analyzed in the presence of small but finite collisional damping. It is shown that there exists a critical collisionality, below which finite amplitude turbulent state can be sustained even in the absence of linear instability. This gives a crude estimate on how “collisionless” is collisionless for nonlinear dynamics of phase space structures to appear.

One of the interesting features of plasmas is that the degrees of freedom in velocity space are important to understand the behavior of plasmas. While wave-particle interaction [1] excites various instabilities, resonance can be strong enough to produce various phase space structures, such as BGK vortex [2], enhanced particle correlations [3, 4], granulations [5], etc. In particular, these phase space structures need not satisfy the dispersion relation, and impact plasma dynamics even when linear modes are stable [6, 7]. Phase space structures are also of practical importance, since they impact the confinement property of fusion plasmas [8–10]. Drift holes, for example, can drive subcritical instability [9] and zonal flows [11]. They can also impact transport by dynamical friction. An impact of phase space structures on the dynamics of magnetically confined plasmas is indicated through the observation of the chirping and nonlinear growth of GAMs [12]. While there are some hints on the role of phase space structures on turbulence and transport, this is still an ongoing topic and remains to be further validated in experiments.

While phase space structures are relevant in understanding the dynamics of turbulent plasmas [13, 14], a natural question arises as to when these effects become relevant. Of course we expect these are important when plasmas are collisionless. Then how collisonless is “collisionless” for such effects to be relevant? Indeed, a small, but finite collision damps a BGK solution eventually [15, 16]. The purpose of this work is to address the question of how collisional effect intervenes the subcritical nature of magnetized plasmas. An explicit condition on the collisionality is derived when such effects become appreciable.

In this work, we use a simplified model to demonstrate the role of collisional dissipation on the subcritical dynamics driven by phase space structures. We focus on the subcritical dynamics driven by a drift hole [9]. In this case, the model is given by

  

$$\frac{dI}{dt}=-C{\nu }_e{I}^{1/4}+{\gamma }_1{I}^{3/2}-\mathrm{\Delta }\omega I^2.$$(1)

Here $I$ is turbulence intensity and given by the mean squared fluctuation amplitude, e.g. $⟨{(e\tilde{\varphi }/T_e)}^2⟩$. The three terms in the RHS of Eq. (1) are explained as follows. The first term describes the nonlinear collisional damping for BGK structure [15, 16]. Here the coefficient $C$ is given as

  

$$C=\frac{7\pi +6}{2\sqrt{\pi }}{\widehat{v}}_{ph}^4\exp (-{\widehat{v}}_{ph}^2/2),$$(2)

where ${\widehat{v}}_{ph}=\omega /(k_{\parallel }{v}_{the})$. This term was originally proposed in literature [15] and numerically verified [16]. While this is derived for BGK vortex in unmagnetized plasmas, we apply this to a drift hole dissipation, since a drift hole has a BGK like structure in the phase space for the parallel dynamics. The second term is related to the nonlinear growth of a drift hole [9, 13]. ${\gamma }_1$ is given as ${\gamma }_1\cong k_{⟂}^2{\rho }_s^{2}f{\omega }_{ci}$, where $f\cong |\mathrm{Im}{\chi }_e\mathrm{Im}{\chi }_i|/{|\chi |}^2$. Here ${\chi }_{e,i}$ is the susceptibility of electrons/ions, respectively, and $\chi ={\chi }_e-{\chi }_i$. Note that $\chi (\omega ,\mathbf{k})$ ≠ 0 since we are interested in the incoherent component of turbulent fluctuations. We also note that $f$ is sensitive to the isotropy of fluctuations in the perpendicular plane to the magnetic field. $f\sim O(1)$ for $k_x\sim k_y$, i.e. when poloidal and radial wave numbers are comparable [13]. The third term is nonlinear damping for saturation [17], and given as $\mathrm{\Delta }\omega \cong {(k_{⟂}^2{\rho }_s^2)}^2{\omega }_{ci}^2/{\gamma }_0$ where ${\gamma }_0$ is in the order of linear growth rate. We note that a generic model for the evolution of $I$ usually contains the linear growth and the nonlinear damping. In contrast, the model used here is specifically for a linearly stable case, so the linear growth is not included. This allows us to focus on the nonlinear dynamics and the effect of finite collision.

We normalize the equation to simplify the analysis. Using the typical amplitude $I_0={({\gamma }_1/\mathrm{\Delta }\omega )}^2$ and the time scale $\tau ={(I_0\mathrm{\Delta }\omega )}^{-1}=\mathrm{\Delta }\omega /{\gamma }_1^2$, the model reduces to

  

$$\frac{dY}{d\widehat{t}}=-\alpha Y^{1/4}+Y^{3/2}-Y^2,$$(3)

and

  

$$\alpha =C\frac{\nu }{\mathrm{\Delta }\omega }{\left(\frac{\mathrm{\Delta }\omega }{{\gamma }_1}\right)}^{7/2}.$$(4)

Here $Y=I/I_0$ is the normalized fluctuation amplitude and $\widehat{t}$ is the dimensionless time. To simplify the notation, the hat is dropped hereafter. After the normalization, the model is characterized by one parameter of collisional damping.

We discuss whether a turbulent state of finite amplitude can be sustained in the presence of collisional dissipation. At the steady state, the local balance yields

  

$$-\alpha Y^{1/4}+Y^{3/2}-Y^2=0.$$(5)

A trivial solution $Y$ = 0 corresponds to no turbulent fluctuation. Apart from this, the local solution needs to satisfy

  

$$\alpha =Y^{5/4}-Y^{7/4}.$$(6)

As shown in Fig. 1, finite amplitude states can be realized for certain collisionality regimes. In particular, the finite amplitude solution exists, in the local model, for

  

$$\alpha <{\alpha }_0\cong 0.1232.$$(7)

$Y$ degenerates at $\alpha ={\alpha }_0$ into $Y_0={(5/7)}^2$. Above this, the nonlinear growth cannot compete against damping, and there is no turbulent state. Thus ${\alpha }_0$ sets an upper bound on the collisionality for turbulent state to be sustained. Finally, we note that there are multiple solutions when finite turbulence exists. Among these, the solution with higher amplitude is likely selected, as discussed in terms of stability with Lyapunov function [13].

Fig. 1.  The dependence of finite amplitude solution on collisionality parameter. Collisionality needs to be small enough for a finite turbulent state to be sustained.

The condition for collisionality is discussed in physical unites here. Restoring the dimensionality, we have

  

$$C\frac{{\nu }_e}{\mathrm{\Delta }\omega }<{\alpha }_0{\left(\frac{{\gamma }_1}{\mathrm{\Delta }\omega }\right)}^{7/2},$$(8)

or

  

$$\frac{{\nu }_e}{c_s/L_n}<\frac{{\alpha }_0}{C}\frac{f^{7/2}}{{(k_{⟂}^2{\rho }_s^2)}^{3/2}}{\left(\frac{{\gamma }_0}{c_s/L_n}\right)}^{5/2}{\left(\frac{c_s/L_n}{{\omega }_{ci}}\right)}^{3/2}.$$(9)

For typical parameters, $k_{⟂}{\rho }_s\sim O(1)$, ${\gamma }_0/(c_s/L_n)\sim \mathrm{Im}{\chi }_e\sim {\widehat{v}}_{ph}$ ∼ 1/4 (i.e. the phase shift is evaluated by electron transit damping [13]), $C\sim 0.03$, $(c_s/L_n)/{\omega }_{ci}={\rho }_s/L_n\sim {10}^{-2}$, $f\sim O(1)$, we have

  

$$\frac{{\nu }_e}{c_s/L_n}<O({10}^{-2})-O({10}^{-3}).$$(10)

This can be met in the core of hot fusion plasmas, such as $T_e$ ∼ 5 keV, $n\sim {10}^{19}{\mathrm{m}}^{-3}$, where $c_s/L_n/2\pi$ ∼ 200 kHz, ${\nu }_e$ ∼ 1.6 kHz, which gives ${\nu }_e/(c_s/L_n)\sim O({10}^{-3})$. Here we note that this is given for illustration under several assumptions. In particular, $f\sim O(1)$ is used by assuming turbulence is isotropic and $\mathrm{Im}{\chi }_e$ is set by electron transit resonance, i.e. independent of collision. Thus this estimate should be taken as a back-of-the-envelope estimate.

In this work, we have investigated how finite collisionality modifies nonlinear dynamics driven by phase-space structures in magnetized plasmas. Using a reduced model based on the dynamics of a drift hole, we discussed how nonlinear collisional damping competes with subcritical excitation by BGK type structures. The analysis demonstrates that collisions do not merely introduce gradual dissipation, but instead impose a well-defined threshold: above a critical collisionality, finite amplitude turbulent states cannot be sustained.

The authors thank P.H. Diamond, T.S. Hahm, G. Dif-Pradalier, K. Ida, and S.-I. Itoh for useful discussion. This work is in part supported by the Grants-in-Aid for Scientific Research of JSPS of Japan (JP21H01066, JP23K20838, JP24K06997, JP26K22311), the joint research project in RIAM, Kyushu University, the NIFS Collaboration Research Program (NIFS26KSPS007), ZE Research Program, IAE, Kyoto University (ZE2026B-12).

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